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Abstract: Let ▫$\mathbb{F}$▫ be a field and ▫$n \ge 3$▫. Suppose ▫${\mathfrak{G_1,G_2}} \subseteq M_n(\mathbb{F})▫$ contain all rank-one idempotents. The structure of surjections ▫$\phi : \mathfrak{G_1} \to \mathfrak{G_2}$▫ satisfying ▫$ABA = 0 \iff \phi(A)\phi(B)\phi(A) = 0$▫ is determined. Similar results are also obtained for (a) subsets of bounded operators acting on a complex or real Banach space, (b) the space of Hermitian matrices acting on ▫$n$▫-dimensional vectors over a skew-field, (c) subsets of self-adjoint bounded linear operators acting on an infinite dimensional complex Hilbert space. It is then illustrated that the results can be applied to characterize mappings ▫$\phi$▫ on matrices or operators such that ▫$F(ABA) = F(\phi(A)\phi(B)\phi(A))▫$ for all ▫$A,B$▫ for functions ▫$F$▫ such as the spectral norm, Schatten ▫$p$▫-norm, numerical radius and numerical range, etc.
Keywords: matrix algebra, Jordan triple product, nonlinear preservers
Published in RUP: 03.04.2017; Views: 2429; Downloads: 97
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Abstract: Let ▫$R$▫ be a 2-torsion free semiprime ▫$\ast$▫-ring and let ▫$\alpha, \beta$▫ be surjective endomorphisms of ▫$R$▫. The aim of the paper is to show that every generalized Jordan triple ▫$(\alpha, \beta)^\ast$▫-derivation on ▫$R$▫ is a generalized Jordan ▫$(\alpha, \beta)^\ast$▫-derivation. This result makes it possible to prove that every generalized Jordan triple ▫$(\alpha, \beta)^\ast$▫-derivation on a semisimple ▫$H^\ast$▫-algebra is a generalized Jordan ▫$(\alpha, \beta)^\ast$▫-derivation. Finally, we prove that every Jordan triple left ▫$\alpha^\ast$▫-centralizer on a 2-torsion free semiprime ring is a Jordan left ▫$\alpha^\ast$▫-centralizer.
Keywords: mathematics, algebra, semiprime ▫$\ast$▫-ring, ▫$H^\ast$▫-algebra, Jordan triple ▫$(\alpha, \beta)^\ast$▫-derivation, generalized Jordan triple ▫$(\alpha, \beta)^\ast$▫-derivation, Jordan triple left ▫$\alpha^\ast$▫-centralizer
Published in RUP: 15.10.2013; Views: 4411; Downloads: 77
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